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Section1.2Vector Equations and Spans

Objectives
  1. Understand the equivalence between a system of linear equations and a vector equation.
  2. Learn the definition of Span { x 1 , x 2 ,..., x k } , and how to draw pictures of spans.
  3. Pictures: an inconsistent system of equations, a consistent system of equations, spans in R 2 and R 3 .
  4. Vocabulary: vector equation.
  5. Essential vocabulary: span.

Subsection1.2.1Vector Equations

An equation involving vectors with n entries is the same as n equations involving only numbers. For example, the equation

x C 126 D + y C − 1 − 2 − 1 D = C 8163 D (1.2.1)

simplifies to

C x 2 x 6 x D + C − y − 2 y − y D = C 8163 D or C x − y 2 x − 2 y 6 x − y D = C 8163 D .

For two vectors to be equal, all of their coordinates must be equal, so this is just the system of linear equations

E x − y = 82 x − 2 y = 166 x − y = 3.
Definition

A vector equation is an equation involving a linear combination of vectors with possibly unknown coefficients.

Asking whether or not a vector equation has a solution is the same as asking if a given vector is a linear combination of some other given vectors.

For example the vector equation above is asking if the vector ( 8,16,3 ) is a linear combination of the vectors ( 1,2,6 ) and ( − 1, − 2, − 1 ) .

Example
Example
Example

Consider the vector equation

x C 111 D + y C 0 − 1 − 1 D + z C 200 D = C 131 D .

Does this vector equation have a solution? Does it have more than one solution?

This vector equation does not have any solution. We could write it as three separate equations:

x + 2 z = 1, x − y = 3, x − y = 1,

which all have to be true at the same time for a solution x , y , z . The second equation says x − y = 3 and the third says x − y = 1. These two statements cannot be true at the same time, no matter what values of x and y we choose. So there can be no solution to the vector equation.

Definition

If one or more solutions exist for an equation or a system of equations, it is said to be consistent. If an equation or system of equations does not have any solution, it is said to be inconsistent.

The above definition is the first of several essential definitions that we will see in this textbook. They are essential in that they form the essence of the subject of linear algebra: learning linear algebra means (in part) learning these definitions. All of the definitions are important, but it is essential that you learn and understand the definitions marked as such.

A Picture of a Consistent System

Below we will show that the above system of equations is consistent. Equivalently, this means that the above vector equation has a solution. In other words, there is a linear combination of ( 1,2,6 ) and ( − 1,2, − 1 ) that equals ( 8,16,3 ) . We can visualize the last statement geometrically. Therefore, the following figure gives a picture of a consistent system of equations. Compare with figure below, which shows a picture of an inconsistent system.

Figure8A picture of the above vector equation. Try to solve the equation geometrically by moving the sliders.

Subsection1.2.2Spans

It will be useful to know what are all linear combinations of a set of vectors v 1 , v 2 ,..., v k in R n . In other words, we would like to understand the set of all vectors b in R n such that the vector equation (in the unknowns x 1 , x 2 ,..., x k )

x 1 v 1 + x 2 v 2 + ··· + x k v k = b

has a solution (i.e. is consistent).

Definition

Let v 1 , v 2 ,..., v k be vectors in R n . The span of v 1 , v 2 ,..., v k is the collection of all linear combinations of v 1 , v 2 ,..., v k , and is denoted

Span { v 1 , v 2 ,..., v k } .

In symbols:

Span { v 1 , v 2 ,..., v k } = A x 1 v 1 + x 2 v 2 + ··· + x k v k | x 1 , x 2 ,..., x k in R B

We also say that Span { v 1 , v 2 ,..., v k } is the subset spanned by or generated by the vectors v 1 , v 2 ,..., v k .

Set Builder Notation

You should read the notation

A x 1 v 1 + x 2 v 2 + ··· + x k v k | x 1 , x 2 ,..., x k in R B

as: “the set of all things of the form x 1 v 1 + x 2 v 2 + ··· + x k v k such that x 1 , x 2 ,..., x k are in R . ” The vertical line is “such that”; everything to the left of it is “the set of all things of this form”, and everything to the right is the condition that those things must satisfy to be in the set. Specifying a set in this way is called set builder notation.

All mathematical notation is only shorthand: any sequence of symbols must translate into an ordinary sentence.

Note(Consistency and span)

Here are two ways of saying the same thing:

  1. A vector b is in the span of v 1 , v 2 ,..., v k .
  2. The vector equation
    x 1 v 1 + x 2 v 2 + ··· + x k v k = b
    is consistent, i.e., it has at least one solution.

Later, in subsection in Section 2.2 we will develop a procedure for answering the question “is b in the span of v 1 , v 2 ,..., v k ? ” in every circumstance. This will be a byproduct of developing a process to find all the solutions of vector equations.

Figure12This is a picture of an inconsistent linear system: the vector w on the right-hand side of the equation x 1 v 1 + x 2 v 2 = w is not in the span of v 1 , v 2 . Convince yourself of this by trying to solve the equation x 1 v 1 + x 2 v 2 = w by moving the sliders, and by row reduction. Compare this figure.
Example
Example
Pictures of Spans

Drawing a picture of Span { v 1 , v 2 ,..., v k } is the same as drawing a picture of all linear combinations of v 1 , v 2 ,..., v k .

Span { v } v Span { v , w } v w Span { v , w } v w
Figure15Pictures of spans in R 2 .
Span { v } v Span { v , w } v w v w u Span { u , v , w } Span { u , v , w } v w u
Figure16Pictures of spans in R 3 . The span of two noncollinear vectors is the plane containing the origin and the heads of the vectors. Note that three coplanar (but not collinear) vectors span a plane and not a 3-space, just as two collinear vectors span a line and not a plane.
Interactive: Span of two vectors in R 2
Interactive: Span of two vectors in R 3
Interactive: Span of three vectors in R 3